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Chapter 1: Problem 27

Evaluate the product, and write the result in the form \(a+b i\) $$4(-1+2 i)$$

Short Answer

Expert verified
The product is \(-4 + 8i\).

Step by step solution

01

Distribute the Real Number

To find the product of a real number with a complex number, distribute the real number to each part of the complex number. In this case, distribute 4 to both \(-1\) and \(2i\).
02

Multiply Each Component Separately

First, multiply 4 by \(-1\) to get:\[4 \times (-1) = -4\]Next, multiply 4 by \(2i\) to get:\[4 \times 2i = 8i\]
03

Combine the Results

Now, combine the results of the multiplication:\[-4 + 8i\]Thus, the product in the form \(a + bi\) is \(-4 + 8i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers are fundamental in mathematics. They include all numbers you can find on the number line.
These include:
  • Whole numbers (like 0, 1, 2)
  • Negative numbers (like -1, -2)
  • Fractions and decimals (like 3/4, 0.25)
Real numbers are used in everyday mathematics and form the backbone for more complex number systems, like imaginary and complex numbers.
In the problem, the number 4 is a real number. When dealing with the multiplication of a real number and a complex number, each part of the complex number is treated as a separate component to be multiplied by the real number.
Distributive Property
The distributive property is a crucial mathematical concept that allows you to multiply a single term by each term inside a bracket in an expression.
In simpler terms, it's a way to "distribute" the number across the terms inside the parentheses. The mathematical rule is expressed as:\[a(b + c) = ab + ac\]Here, the real number 4 is multiplied by each term inside the parentheses i.e., -1 and 2i.
This method simplifies the process of dealing with equations involving terms within brackets or parentheses.
  • You multiply the real number (like 4) with each component of the complex number separately.
  • This ensures every part of the expression is correctly factored by the outside term.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of negative one:\[ i = \sqrt{-1} \]This seems strange because no real number multiplied by itself will give a negative result.
That's what makes imaginary numbers fascinating and essential in advanced mathematics.
In our original problem, the imaginary part is represented by "2i."
  • When multiplied by any real number, it retains its imaginary nature, but the coefficient changes.
  • For example, in the solution, 4 is multiplied by 2i to get 8i.
Understanding \(i\) helps bridge the gap between real number arithmetic and more advanced calculations involving complex numbers.

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